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Theorem fin23lem30 9752
Description: Lemma for fin23 9799. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem30 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . 2 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4503 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 217 . 2 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
4 simpr 485 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5 fin23lem.d . . . . . . . . . . . 12 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
65funmpt2 6387 . . . . . . . . . . 11 Fun 𝑅
7 funco 6388 . . . . . . . . . . 11 ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡𝑅))
84, 6, 7sylancl 586 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡𝑅))
9 elunirn 7001 . . . . . . . . . 10 (Fun (𝑡𝑅) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
108, 9syl 17 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
11 dmcoss 5835 . . . . . . . . . . . 12 dom (𝑡𝑅) ⊆ dom 𝑅
1211sseli 3960 . . . . . . . . . . 11 (𝑏 ∈ dom (𝑡𝑅) → 𝑏 ∈ dom 𝑅)
13 fvco 6752 . . . . . . . . . . . . . . . 16 ((Fun 𝑅𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
146, 13mpan 686 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑅 → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1514adantl 482 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1615eleq2d 2895 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅𝑏))))
17 incom 4175 . . . . . . . . . . . . . . . 16 ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏)))
18 difss 4105 . . . . . . . . . . . . . . . . . . . . . . 23 (ω ∖ 𝑃) ⊆ ω
19 ominf 8718 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ ω ∈ Fin
20 fin23lem.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
2120ssrab3 4054 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑃 ⊆ ω
22 undif 4426 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
2321, 22mpbi 231 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 ∪ (ω ∖ 𝑃)) = ω
24 unfi 8773 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
2523, 24eqeltrrid 2915 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
2625ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
2719, 26mtoi 200 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
2827ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
295fin23lem22 9737 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
3018, 28, 29sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
31 f1of 6608 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
3230, 31syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
33 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
3432fdmd 6516 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
3533, 34eleqtrd 2912 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
3632, 35ffvelrnd 6844 . . . . . . . . . . . . . . . . . . . 20 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ (ω ∖ 𝑃))
3736eldifbd 3946 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ 𝑃)
3820eleq2i 2901 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ 𝑃 ↔ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
3937, 38sylnib 329 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4036eldifad 3945 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ ω)
41 fveq2 6663 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑅𝑏) → (𝑡𝑣) = (𝑡‘(𝑅𝑏)))
4241sseq2d 3996 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑅𝑏) → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4342elrab3 3678 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ ω → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4440, 43syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4539, 44mtbid 325 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)))
46 fin23lem.a . . . . . . . . . . . . . . . . . . 19 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
4746fin23lem20 9747 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑏) ∈ ω → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
4840, 47syl 17 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
49 orel1 882 . . . . . . . . . . . . . . . . 17 ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) → (( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
5045, 48, 49sylc 65 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅)
5117, 50syl5eq 2865 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅)
52 disj 4395 . . . . . . . . . . . . . . 15 (((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
5351, 52sylib 219 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
54 rsp 3202 . . . . . . . . . . . . . 14 (∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5553, 54syl 17 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5616, 55sylbid 241 . . . . . . . . . . . 12 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
5756ex 413 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
5812, 57syl5 34 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
5958rexlimdv 3280 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
6010, 59sylbid 241 . . . . . . . 8 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) → ¬ 𝑎 ran 𝑈))
6160ralrimiv 3178 . . . . . . 7 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
62 disj 4395 . . . . . . 7 (( ran (𝑡𝑅) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
6361, 62sylibr 235 . . . . . 6 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅)
64 rneq 5799 . . . . . . . . 9 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6564unieqd 4840 . . . . . . . 8 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6665ineq1d 4185 . . . . . . 7 (𝑍 = (𝑡𝑅) → ( ran 𝑍 ran 𝑈) = ( ran (𝑡𝑅) ∩ ran 𝑈))
6766eqeq1d 2820 . . . . . 6 (𝑍 = (𝑡𝑅) → (( ran 𝑍 ran 𝑈) = ∅ ↔ ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅))
6863, 67syl5ibr 247 . . . . 5 (𝑍 = (𝑡𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran 𝑍 ran 𝑈) = ∅))
6968expd 416 . . . 4 (𝑍 = (𝑡𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)))
7069impcom 408 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
71 rneq 5799 . . . . . . . 8 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7271unieqd 4840 . . . . . . 7 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7372ineq1d 4185 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈))
74 rncoss 5836 . . . . . . . 8 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
7574unissi 4853 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
76 disj 4395 . . . . . . . 8 (( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ¬ 𝑎 ran 𝑈)
77 eluniab 4841 . . . . . . . . . 10 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)))
78 eleq2 2898 . . . . . . . . . . . . . 14 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈)))
79 eldifn 4101 . . . . . . . . . . . . . 14 (𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈) → ¬ 𝑎 ran 𝑈)
8078, 79syl6bi 254 . . . . . . . . . . . . 13 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8180rexlimivw 3279 . . . . . . . . . . . 12 (∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8281impcom 408 . . . . . . . . . . 11 ((𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8382exlimiv 1922 . . . . . . . . . 10 (∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8477, 83sylbi 218 . . . . . . . . 9 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} → ¬ 𝑎 ran 𝑈)
85 eqid 2818 . . . . . . . . . . 11 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
8685rnmpt 5820 . . . . . . . . . 10 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8786unieqi 4839 . . . . . . . . 9 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8884, 87eleq2s 2928 . . . . . . . 8 (𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8976, 88mprgbir 3150 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅
90 ssdisj 4405 . . . . . . 7 (( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∧ ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅) → ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅)
9175, 89, 90mp2an 688 . . . . . 6 ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅
9273, 91syl6eq 2869 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ∅)
9392a1d 25 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9493adantl 482 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9570, 94jaoi 851 . 2 (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
961, 3, 95mp2b 10 1 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  ifcif 4463  𝒫 cpw 4535   cuni 4830   cint 4867   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  ccom 5552  suc csuc 6186  Fun wfun 6342  wf 6344  1-1-ontowf1o 6347  cfv 6348  crio 7102  (class class class)co 7145  cmpo 7147  ωcom 7569  seqωcseqom 8072  m cmap 8395  cen 8494  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seqom 8073  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356
This theorem is referenced by:  fin23lem31  9753
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